Triangle Center

A triangle center is a point whose Trilinear Coordinates are defined in terms of the side lengths and angles of a Triangle. The function giving the coordinates $\alpha:\beta:\gamma$ is called the Triangle Center Function. The four ancient centers are the Centroid, Incenter, Circumcenter, and Orthocenter. For a listing of these and other triangle centers, see Kimberling (1994).

A triangle center is said to be Regular Iff there is a Triangle Center Function which is a Polynomial in $\Delta$, $a$, $b$, and $c$ (where $\Delta$ is the Area of the Triangle) such that the Trilinear Coordinates of the center are

$$f(a,b,c):f(b,c,a):f(c,a,b).$$

A triangle center is said to be a Major Triangle Center if the Triangle Center Function $\alpha$ is a function of Angle $A$ alone, and therefore $\beta$ and $\gamma$ of $B$ and $C$ alone, respectively.

See also Major Triangle Center, Regular Triangle Center, Triangle, Triangle Center Function, Trilinear Coordinates, Trilinear Polar

References

Davis, P. J. ``The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History.'' Amer. Math. Monthly 102, 204-214, 1995.

Dixon, R. ``The Eight Centres of a Triangle.'' §1.5 in Mathographics. New York: Dover, pp. 55-61, 1991.

Gale, D. ``From Euclid to Descartes to Mathematica to Oblivion?'' Math. Intell. 14, 68-69, 1992.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-167, 1994.

Kimberling, C. ``Triangle Centers and Central Triangles.'' Congr. Numer. 129, 1-295, 1998.